Find the L.C.M. of the numbers 22, 54, 108, 135 and 198.

Introduction / Context:
This numerical aptitude question tests your ability to find the least common multiple (L.C.M.) of several integers. L.C.M. problems are frequently asked in exams because they are fundamental to understanding multiples, divisibility, and synchronizing events or cycles based on different periods.

Given Data / Assumptions:

• The numbers given are 22, 54, 108, 135, and 198.
• We need to compute the least common multiple of these numbers.
• All numbers are positive integers.

Concept / Approach:
The L.C.M. of a set of numbers is the smallest positive integer that is divisible by each of the numbers. The most systematic way to find the L.C.M. is to use prime factorization. We factor each number into primes, then take the highest power of every prime that appears in any factorization. Multiplying these highest powers gives the L.C.M.

Step-by-Step Solution:
Step 1: Prime factorize each number.22 = 2 * 11.54 = 2 * 3^3.108 = 2^2 * 3^3.135 = 3^3 * 5.198 = 2 * 3^2 * 11.Step 2: Identify all distinct prime factors: 2, 3, 5, and 11.Step 3: Take the highest power of each prime that appears: 2^2 (from 108), 3^3 (from 54, 108, 135), 5^1 (from 135), and 11^1 (from 22 or 198).Step 4: Multiply these highest powers: L.C.M. = 2^2 * 3^3 * 5 * 11.Step 5: Compute: 2^2 = 4; 3^3 = 27. So L.C.M. = 4 * 27 * 5 * 11.Step 6: 4 * 27 = 108; 108 * 5 = 540; 540 * 11 = 5940.Step 7: Therefore, the L.C.M. is 5940.

Verification / Alternative check:
Check divisibility: 5940 ÷ 22, 5940 ÷ 54, 5940 ÷ 108, 5940 ÷ 135, and 5940 ÷ 198 all give integer results. Since all the original numbers divide 5940 and no smaller number containing all required prime powers exists, 5940 is indeed the least common multiple.

Why Other Options Are Wrong:
Option a (330) and option b (1980) are too small; they fail to be divisible by all of the given numbers, especially 108 and 135. Option d (11880) is a common multiple but not the least one; it is simply 2 * 5940. Only 5940 is the smallest integer divisible by each of the five numbers.

Common Pitfalls:
Many students incorrectly combine factors or forget to take the highest power of each prime, which leads to a number that is not divisible by all given values. Another common error is to try to find the L.C.M. pairwise and to accumulate mistakes. Prime factorization keeps the process systematic and reduces the chance of missing any prime power.

Final Answer:
The L.C.M. of 22, 54, 108, 135, and 198 is 5940.